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We consider a class of rigidity results in a convex cone $\Sigma \subseteq\mathbb{R}^N$. These include overdetermined Serrin-type problems for a mixedboundary value problem relative to $\Sigma$, Alexandrov's soap bubble-typeresults relative to $\Sigma$, and a Heintze-Karcher's inequality relative to$\Sigma$. Each rigidity result is obtained by means of a single integralidentity and holds true under weak integral conditions. Optimal quantitativestability estimates are obtained in terms of an $L^2$-pseudodistance. Inparticular, the optimal stability estimate for Heintze-Karcher's inequality isnew even in the classical case $\Sigma = \mathbb{R}^N$.  Stability bounds in terms of the Hausdorff distance are also provided.  Several new results are established and exploited, including a newPoincar\'e-type inequality for vector fields whose normal components vanish ona portion of the boundary and an explicit (possibly weighted) trace theory --relative to the cone $\Sigma$ -- for harmonic functions satisfying ahomogeneous Neumann condition on the portion of the boundary contained in$\partial \Sigma$.  We also introduce new notions of uniform interior and exterior sphereconditions relative to the cone $\Sigma \subseteq \mathbb{R}^N$, which allow toobtain (via barrier arguments) uniform lower and upper bounds for the gradientin the mixed boundary value-setting. In the particular case $\Sigma =\mathbb{R}^N$, these conditions return the classical uniform interior andexterior sphere conditions (together with the associated classical gradientbounds of the Dirichlet setting).

Soap bubbles and convex cones: optimal quantitative rigidity